use the trapezoidal rule, the midpoint rule, and simpson’s rule to approximate the integral ∫₁⁹ (ln(x)/(5 +…

Explanation:

Step1: Calculate $\Delta x$

$\Delta x=\frac{b - a}{n}$, where $a = 1$, $b = 9$, $n=12$. So $\Delta x=\frac{9 - 1}{12}=\frac{2}{3}$.

Step2: Define the function

Let $f(x)=\frac{\ln(x)}{5 + x}$.

Step3: Trapezoidal Rule ($T_{n}$)

$T_{n}=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n - 1})+f(x_n)]$. $x_i=a + i\Delta x$, for $i = 0,1,\cdots,12$. $x_0 = 1,x_1=1+\frac{2}{3}=\frac{5}{3},x_2 = 1 + 2\times\frac{2}{3}=\frac{7}{3},\cdots,x_{12}=9$. $T_{12}=\frac{\frac{2}{3}}{2}[f(1)+2f(\frac{5}{3})+2f(\frac{7}{3})+\cdots+2f(\frac{35}{3})+f(9)]$. $f(1)=\frac{\ln(1)}{5 + 1}=0$. Calculate each term and sum them up: [ \begin{align*} T_{12}&=\frac{1}{3}\left[0 + 2\times\frac{\ln(\frac{5}{3})}{5+\frac{5}{3}}+2\times\frac{\ln(\frac{7}{3})}{5+\frac{7}{3}}+\cdots+2\times\frac{\ln(\frac{35}{3})}{5+\frac{35}{3}}+\frac{\ln(9)}{5 + 9}\right]\ &\approx1.0688 \end{align*} ]

Step4: Mid - point Rule ($M_{n}$)

$M_{n}=\Delta x[f(\overline{x_1})+f(\overline{x_2})+\cdots+f(\overline{x_n})]$, where $\overline{x_i}=\frac{x_{i - 1}+x_i}{2}$. $\overline{x_1}=\frac{1+\frac{5}{3}}{2}=\frac{4}{3},\overline{x_2}=\frac{\frac{5}{3}+\frac{7}{3}}{2}=2,\cdots$. $M_{12}=\frac{2}{3}\left[f(\frac{4}{3})+f(2)+f(\frac{8}{3})+\cdots+f(\frac{26}{3})\right]$. Calculate each term and sum them up: [ \begin{align*} M_{12}&=\frac{2}{3}\left[\frac{\ln(\frac{4}{3})}{5+\frac{4}{3}}+\frac{\ln(2)}{5 + 2}+\frac{\ln(\frac{8}{3})}{5+\frac{8}{3}}+\cdots+\frac{\ln(\frac{26}{3})}{5+\frac{26}{3}}\right]\ &\approx1.0567 \end{align*} ]

Step5: Simpson's Rule ($S_{n}$)

$S_{n}=\frac{\Delta x}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots+2f(x_{n - 2})+4f(x_{n - 1})+f(x_n)]$ for $n$ even. $S_{12}=\frac{\frac{2}{3}}{3}[f(1)+4f(\frac{5}{3})+2f(\frac{7}{3})+4f(\frac{9}{3})+\cdots+2f(\frac{31}{3})+4f(\frac{33}{3})+f(9)]$. Calculate each term and sum them up: [ \begin{align*} S_{12}&=\frac{2}{9}\left[0 + 4\times\frac{\ln(\frac{5}{3})}{5+\frac{5}{3}}+2\times\frac{\ln(\frac{7}{3})}{5+\frac{7}{3}}+4\times\frac{\ln(3)}{5 + 3}+\cdots+2\times\frac{\ln(\frac{31}{3})}{5+\frac{31}{3}}+4\times\frac{\ln(11)}{5 + 11}+\frac{\ln(9)}{5 + 9}\right]\ &\approx1.0632 \end{align*} ]

Answer:

$T_{12}\approx1.0688$ $M_{12}\approx1.0567$ $S_{12}\approx1.0632$